Using GeoGeobra, I constructed the point for a triangle of sides $AB=13$, $BC=6$ and $CA=9$, and computed the (oriented) distance from it to the line $(BC)$, getting a value $kx\approx -13.209$.
Comparing with the list https://faculty.evansville.edu/ck6/encyclopedia/Search_6_9_13.html, I got a match as the point $X(1337)$, that is the "1st Wernau Point", whose definition matches yours. For references (Olympiad problems), see https://faculty.evansville.edu/ck6/encyclopedia/ETCPart2.html